Newton's Bucket

Hi. First post here. I have no formal math or physics training, but read popular books on physics and am pretty well read as far as that goes. Now for the question.

I'm fascinated by the Newton's Bucket problem and fortunately for me it's cleared my head of the 2 brothers paradox (one on earth, one in ship, ship ages) with regard to which one is considered moving and which is stationary.

I've never liked the traditional idea that the brother that is considered moving (and therefore aging) is the one that is accelerating away because once acceleration stops and the ship continues at near light speed, the aging process continues yet the ship is only moving relative to the Earth and not accelerating away from it.

Newton's Bucket solves that problem by inferring that the ship is moving near light speed relative to either the stars or some universal fabric that is static or almost static relative to the stars.

Newton's bucket implies that if the universe were empty (I suppose this would include dark matter and energy) except for the bucket and a single observer, the bucket would seemingly have to behave strangely. For example, if the observer were spinning around the bucket (and the bucket around the observer) but both in the same direction as far as the two axis of rotation are concerned, the bucket could not be said to be spinning and therefore would not exhibit inertial forces or the resultant concave water. If the observer and bucket were spinning opposite to each other, then what? Would the water then become concave relative to the velocity of the observer? Or is a greater mass (or something else altogether) required such as massive galaxies? And if either or both are causing the water to become concave, then what exactly is causing it. I realize the simple answer is inertia, but this paradox implies that inertia would cease to exist in an empty universe and with the observer and bucket moving in the same direction or possibly in different directions as well.

Inertia would have to cease to exist in an empty universe that contained only a bucket of water and a single observer moving in the same direction around it as there would be absolutely no frame of reference with regard to acceleration. With no inertia, one could not feel any effects of acceleration so if the bucket exploded, or the observer sneezed, which would move relative to the other, and which one would age when applied to the two brother paradox.

".....in simple terms, in a universe with no matter there is no gravity. Hence general relativity reduces to special relativity and now all observers agree when the rock system is spinning (i.e. accelerating). "

In other words relativity says rotation is detectable even with one object in an empty universe. Of course this is hard to prove with an experiment, as we do not have a spare empty universe to try it out in :P

Tha article also tries to lend some support to Mach's views (that all inertia is relative to the fixed stars):

"In 1985 further progress by H Pfister and K Braun showed that sufficient centrifugal forces would be induced at the centre of the hollow massive sphere to cause water to form a concave surface in a bucket which is not rotating with respect to the distant stars. Here at last was a form of the symmetry that Mach was seeking. "

A counter argument is this:

Rotate a bucket clockwise (when looking from above) so that the water contained within it has a concave surface. Define the bucket as stationary and atribute the concave surface of the water to the gravitational influence of the all the universes stars orbiting anti-clockwise around said bucket. Now place another rotating bucket alongside the first bucket while the water within it is still spinning. If the first bucket is exactly at the axis of the spinning universe, then the second bucket is not and yet the lowest point of the water in the second bucket is exactly at the centre of its spinning surface. Mach's principle seems to fall apart as soon as we introduce a second bucket.

"Newton's Bucket" only works in the presence of gravity as kev pointed out.

That said - the pressure in the water increases linearly from 0 to [itex] \rho gh[/itex] no matter where you check from top to bottom. When the bucket/water is spinning uniformly, a new force is added to keep the water from travelling along a linear path. This new force creates another linear pressure gradient that starts from the center of the bucket and increases as you move away from the axis of rotation. The product of the two orthogonal linear pressure gradients leads to a parabolic pressure profile at any fixed height. The water surface assumes a parabolic shape to support both linear pressure gradients simultaneously.

".....in simple terms, in a universe with no matter there is no gravity. Hence general relativity reduces to special relativity and now all observers agree when the rock system is spinning (i.e. accelerating). "

In other words relativity says rotation is detectable even with one object in an empty universe. Of course this is hard to prove with an experiment, as we do not have a spare empty universe to try it out in :P

But the article just a little before that quote also states that Einstein said that Mach's view was in complete agreement with GR so this conclusion in the article confused me. I'm also confused about how observers could agree that the bucket is spinning. Because the water would go concave? Again, why would it go concave in an empty universe?

Tha article also tries to lend some support to Mach's views (that all inertia is relative to the fixed stars):

"In 1985 further progress by H Pfister and K Braun showed that sufficient centrifugal forces would be induced at the centre of the hollow massive sphere to cause water to form a concave surface in a bucket which is not rotating with respect to the distant stars. Here at last was a form of the symmetry that Mach was seeking. "

Yes, I was trying not to bring this up too soon, but logically, I'm in agreement with this.

A counter argument is this:

Rotate a bucket clockwise (when looking from above) so that the water contained within it has a concave surface. Define the bucket as stationary and atribute the concave surface of the water to the gravitational influence of the all the universes stars orbiting anti-clockwise around said bucket. Now place another rotating bucket alongside the first bucket while the water within it is still spinning. If the first bucket is exactly at the axis of the spinning universe, then the second bucket is not and yet the lowest point of the water in the second bucket is exactly at the centre of its spinning surface. Mach's principle seems to fall apart as soon as we introduce a second bucket.

If the first bucket were indeed at the very axis of the spinning universe and by definition not spinning, then the concaveness of the water would be due to a force (gravity or otherwise) from the stars pulling equally at all sides of the water causing it to rise up the sides of the bucket. (One could no longer state inertia being the cause as the bucket is "not spinning") A second bucket placed off center would also feel this same "pull" and it's water would also rise, but one side would rise higher then the other, having a stronger "pull" on that side. Because of the scales the offset would be infintesimally small, perhaps a plank length. In a universe with non-rotating stars (the real universe) one cannot say that two spinning buckets side by side have their dips in the absolute center of the bucket or that the two buckets have their dips in the same location.[/QUOTE]

Mach's Principle might not rely on just gravitational influences, as it would in GR.

In the Brans Dicke theory an extra scalar field coupled to matter endows fundamental particles with inertial mass.

Thus introducing the second bucket proves that Mach's Principle is incompatible with GR but it may not be incompatible with an alternative gravitational theory.

Garth

My gut tells me that gravity can't play much of a part in Mach's Principle as the stars are simply too far away. Doesn't gravity eventually diminish to a single planck value at which point gravity can be said to not exist at all? Of course there is still the sun and a spinning bucket in our universe may be under it's sole influence. Is there anyway to determine this or is there any theory indicating this? Also, I am a bit confused by the terms tensor vs scaler. Wikipedia didn't help me much here, can you explain this in simple (non-math) terms?

"Newton's Bucket" only works in the presence of gravity as kev pointed out.

It does seem that gravity is the most suspect reason for Mach's principle, but how about a situation of an empty universe with one bucket of water and one observer. If the observer were to grab the bucket and spin it then one of three things would happen. 1) The water would go noticably concave (and simultaneously the observer would also feel a centrifugal force on it's own body) due to both spinning relative to a (non local) absolute space, 2) the water would stay flat even though it was spinning relative to the observer because there is no absolute space. or 3) there would be an infintesimally small inertial force on both the bucket of water (causing it to go ever so slightly concave) and the observer due to both spinning relative to each other and because the delta between the masses of the two objects define an absolute space that is moving more slowly relative to the more massive object then it is to the less massive object.

It does seem that gravity is the most suspect reason for Mach's principle, but how about a situation of an empty universe with one bucket of water and one observer.

What would keep the water in the bucket ? The water would form into a sphere and freeze. It's been pointed out to you that the parabolic surface is due to a combination of lateral and vertical forces, so talking about the surface of the water in your scenario isn't realistic.

I would expect any spinning object to experience stresses because of the spin, and this would happen in any sort of universe, regardless of gravity.

What would keep the water in the bucket ? The water would form into a sphere and freeze. It's been pointed out to you that the parabolic surface is due to a combination of lateral and vertical forces, so talking about the surface of the water in your scenario isn't realistic.

I was using the bucket in the spirit of a thought experiment for it's ease of visualization. It is a totally impractical object to use in a real experiment, but the point of my original post is that you can use any practical object here with the same effect. For example two spheres tied together with a string and spun around the axis of the center of the string, or an elastic sphere which would bulge at the center and so on. The actual object is not important here, only the fact that there is centrifugal forces acting on that object.

I would expect any spinning object to experience stresses because of the spin, and this would happen in any sort of universe, regardless of gravity.

Not so if Mach's Principle were true. In an empty universe there would be no stresses on a spinning object because there would be way to know what that object was spinning in reference to, or in other words, whether it was spinning at all. This does have the deeper implication that in an empty universe what we know of as inertia would cease to exist altogether. For example, if you were in a spaceship in an empty universe and flipped the switch to start the rocket engine, it would fire (maybe), but there would be no sensation of forward thrust, the accelerometer onboard would not show any change, you would not feel any G force, and in essense Newton's 3 laws of motion would break down.

Please realize though that I am also trying to figure out here what is an "empty universe". Is it simply a universe void of matter? Of dark matter and dark energy? Of virtual particles? Also, I'm not completely convinced that it's matter that is the real reference point for a spinning object and it's associated stresses (acceleration). It could also be that even an empty universe has some kind of inherent frame of reference that defines that it is static and not moving regardless of whether or not it contains matter, dark matter, and/or dark energy. If this is the case, then I would think a spinning object would still show rotational forces acting on it even in a massless universe. But if this is the case, then it would turn the physics world upside down I would think.

Not so if Mach's Principle were true. In an empty universe there would be no stresses on a spinning object because there would be way to know what that object was spinning in reference to, or in other words, whether it was spinning at all. This does have the deeper implication that in an empty universe what we know of as inertia would cease to exist altogether. For example, if you were in a spaceship in an empty universe and flipped the switch to start the rocket engine, it would fire (maybe), but there would be no sensation of forward thrust, the accelerometer onboard would not show any change, you would not feel any G force, and in essense Newton's 3 laws of motion would break down.

OK, from your earlier remarks I can see we are on the same playing field now. I will try and refute the bit I've quoted above.

Firstly, rotation can only be defined for an extended object. A point cannot rotate. So the parts of the extended object have proper spatial relationships with each other and provide a frame in which to define rotation independently of any external reference. I can choose the centre of the rotation as the origin of a frame, and then define a tangential velocity of a piece away from the centre.

The same argument might well do for the acceleration case, but you should bear in mind that your one single object in the universe can only accelerate by ejecting some matter, in which case we have more than one object and the argument short circuits.

[edit] Re-reading this, I'm not 100% convinced by my logic, it would be interesting to hear other views.

The actual object is not important here, only the fact that there is centrifugal forces acting on that object.

"Centrifugal force" is an artificial construct used to balance the centripetal force (i.e. that exerted by the bucket wall) acting on an object that would otherwise travel in a straight line. The closest approximation to a "centrifugal force" would be the tendency of like charges to repel one another - which isn't the sort of thing that keeps a mass rotating at constant radius.

I am aware that Einstein himself concluded that Mach's principle is incompatible with GR as demonstrated by this quote:

"This certainly was a clever idea on Einstein's part, but by June 1918 it had become clear that the De Sitter world does not contain any hidden masses and is thus a genuine counterexample to Mach's principle. Another one of Einstein's attempts to relativize all motion had failed.
Einstein thereupon lost his enthusiasm for Mach's principle. He accepted that motion with respect to the metric field cannot always be translated into motion with respect to other matter."

However, after further reflection Mach's principle is not dismissed by the simple counter example I gave. In that example the second bucket would appear to be rotating along with the distant stars from the point of view of an observer stationary with respect to the water in the first bucket. The second bucket would not therefore be submitted to the "spiralling spacetime" that the water in the first bucket is subjected to, because the second bucket is comoving with the spiralling spacetime/ gravitational field.

A clearer (and fairer) example would be to place the first bucket at the centre of a large rotating turntable. An observer on the turntable could place a second bucket near the rim of the turntable and observe that the water in the second bucket is at rest with with respect to the water of the first bucket and that the water in the second bucket is piled up asymmetrically on the side furthest from the centre of the turntable. If the water in the second bucket is spinning then the centre of the concave depression would indeed be offset from the centre of the bucket. In this fairer second example, Mach's principle does not fail. Can anyone think of a simple example (that is easy to visualise), where Mach's principle fails?

[edit] Re-reading this, I'm not 100% convinced by my logic, it would be interesting to hear other views.

Consider that a fixed volume following a curved path will have different velocities at different points on/within that fixed volume. If the differential velocities become too great, the object flies apart.

The spherical blob of water you mentioned only remains so because of surface tension. If that blob of water were to rotate about some axis, there would have to be more surface area in a plane perpendicular to the axis of rotation to keep the forces in equilibrium - leading to an ellipsoidal shape.

Oddly, a spherical blob of water travelling at a significant fraction of the speed of light would also look like an ellipsoid to a stationary observer - but for a different reason.

It fails on Occams razor, surely. There's nothing to explain. All rotating phenomena are accounted for by present dynamics without need for a cosmic frame. Or am I missing something deep here ?

General Relativity can explain any motion including accelerated motion in a straight line in terms of no motion and and complicated gravitational spacetime. For example, if you turn on your rocket motor and accelerate from a standstill to 0.8c, it can be explained in terms of a gravitational field that springs up the instant you turned your rocket motor on and draws the universe towards a black hole behind you while your rocket motor resists the gravitational "pull".

When you drive to work, accelerating and breaking at junctions and experiencing "centrifugal force" as you go round corners, the whole journey can be explained in terms of gravitational fields and complicated accelerations of everything in the universe while you have remained stationary throughout the entire journey. Now this point of view is necessary or we have to accept a notion of absolute motion which is incompatible with Relativity. Occam's razor and even considerations of conservation of energy are not strong enough arguments to support a notion of absolute motion or acceleration.

Firstly, rotation can only be defined for an extended object. A point cannot rotate. So the parts of the extended object have proper spatial relationships with each other and provide a frame in which to define rotation independently of any external reference. I can choose the centre of the rotation as the origin of a frame, and then define a tangential velocity of a piece away from the centre.

Let's take two bricks tied together by a rope and define that the bricks are not spinning (one face of each brick always faces the other). If there is tension on the rope, then one can say the bricks are revolving about each other. But in an empty universe, this would mean the system would be revolving relative to absolute space. If there is no absolute space, then there could be no tension on the rope since the objects are not rotating relative to anything (not even to each other if their faces are stationary)

The same argument might well do for the acceleration case, but you should bear in mind that your one single object in the universe can only accelerate by ejecting some matter, in which case we have more than one object and the argument short circuits.

The rocket is a matter/anti-matter engine and all exhaust is converted into energy.

Now this point of view is necessary or we have to accept a notion of absolute motion which is incompatible with Relativity.

Well, I don't see at all how that follows from your argument. I can accept absolute rotation, because of the extended object argument, and I think acceleration can always be detected so it's got nothing to do with absolute motion.

But the universe is not empty, it has a rope and two bricks in it ! It's like saying 'take a full, empty glass of water ...'.
If I define a frame centred on one brick, the other is rotating around it !

The only way you would know that one was rotating around the other would be if the rope were taught. If the rope were limp, then you could conclude one brick was not rotating around the other, but this simply takes the argument back to the beginning of the Newton's bucket problem in the first place. The problem is not determining if there is revolution by looking at the rope. This is a given. The problem is why is the rope taught or limp in the first place when there is no way to determine (in an empty universe) if the objects are revolving around each other. In a universe with no absolute space (or space-time) there is no frame of reference to determine whether a rope should be limp or taught. In other words, if the rope is taught in an empty universe, then this is irrefutable evidence that there is a static frame of reference that is not rotating relative to the rotating objects.

This static frame of reference can be absolute space (Newton's absolute space or Minkowski's absolute space-time) or it could be the total relative position of the stars (Mach's principle) that is the cause of the taught rope. But it has to be one or the other from what I can see. If neither was the cause the rope could never become taught.

I just had a thought: If indeed the culprit were absolute space and not Mach's principle, could the stars be revolving slowly with respect to this absolute space and therfore have an outward inertial force on them causing the universe to accelerate apart? In other words could this explain the accelerating expanding universe without resorting to dark energy (or Einsteins cosmological constant) to explain this expansion?

I just had a thought: If indeed the culprit were absolute space and not Mach's principle, could the stars be revolving slowly with respect to this absolute space and therfore have an outward inertial force on them causing the universe to accelerate apart? In other words could this explain the accelerating expanding universe without resorting to dark energy (or Einsteins cosmological constant) to explain this expansion?

The difficulty with using rotation in explaining the expansion of universe is that the expansion would only occur around the "equator" of the universe and not at the "poles" alligned with the rotation. I don't think it is possible to rotate a sphere about 3 axes simultaneously so that "centrifugal force" appears to act equally in all directions.